gausslemma2d - Metamath Proof Explorer

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Theorem gausslemma2d 26050

Description: Gauss' Lemma (see also theorem 9.6 in [ApostolNT] p. 182) for integer 2: Let p be an odd prime. Let S={2,4,6,...,(p-1)}. Let n denote the number of elements of S whose least positive residue modulo p is greater than p/2. Then ( 2 | p ) = (-1)^n. (Contributed by AV, 14-Jul-2021.)

**Hypotheses**
Ref	Expression
gausslemma2d.p	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
gausslemma2d.h	⊢ 𝐻 = ((𝑃 − 1) / 2)
gausslemma2d.r	⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
gausslemma2d.m	⊢ 𝑀 = (⌊‘(𝑃 / 4))
gausslemma2d.n	⊢ 𝑁 = (𝐻 − 𝑀)

**Assertion**
Ref	Expression
gausslemma2d	⊢ (𝜑 → (2 /_L 𝑃) = (-1↑𝑁))

Distinct variable groups: 𝑥,𝐻 𝑥,𝑃 𝜑,𝑥 𝑥,𝑀 Allowed substitution hints: 𝑅(𝑥) 𝑁(𝑥)

**Proof of Theorem gausslemma2d**
Step	Hyp	Ref	Expression
1		gausslemma2d.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
2		gausslemma2d.h	. . 3 ⊢ 𝐻 = ((𝑃 − 1) / 2)
3		gausslemma2d.r	. . 3 ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
4		gausslemma2d.m	. . 3 ⊢ 𝑀 = (⌊‘(𝑃 / 4))
5		gausslemma2d.n	. . 3 ⊢ 𝑁 = (𝐻 − 𝑀)
6	1, 2, 3, 4, 5	gausslemma2dlem7 26049	. 2 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)
7		eldifi 4033	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
8		prmnn 16063	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	8	nnred 11682	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
10		prmgt1 16086	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
11	9, 10	jca 516	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
12	1, 7, 11	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
13		1mod 13313	. . . . . 6 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → (1 mod 𝑃) = 1)
15	14	eqcomd 2765	. . . 4 ⊢ (𝜑 → 1 = (1 mod 𝑃))
16	15	eqeq2d 2770	. . 3 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1 ↔ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)))
17		neg1z 12050	. . . . . . . . . . 11 ⊢ -1 ∈ ℤ
18	1, 4, 2, 5	gausslemma2dlem0h 26039	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
19		zexpcl 13487	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℤ)
20	17, 18, 19	sylancr 591	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ)
21		2nn 11740	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℕ)
23	1, 2	gausslemma2dlem0b 26033	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ ℕ)
24	23	nnnn0d 11987	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
25	22, 24	nnexpcld 13649	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑𝐻) ∈ ℕ)
26	25	nnzd 12118	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ)
27	20, 26	zmulcld 12125	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ)
28	27	zred 12119	. . . . . . . 8 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℝ)
29		1red 10673	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
30	28, 29	jca 516	. . . . . . 7 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ))
31	30	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ))
32	1	gausslemma2dlem0a 26032	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
33	32	nnrpd 12463	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
34	20, 33	jca 516	. . . . . . 7 ⊢ (𝜑 → ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺))
35	34	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺))
36		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃))
37		modmul1 13334	. . . . . 6 ⊢ (((((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ) ∧ ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺) ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃))
38	31, 35, 36, 37	syl3anc 1369	. . . . 5 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃))
39	38	ex 417	. . . 4 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃)))
40	20	zcnd 12120	. . . . . . . . 9 ⊢ (𝜑 → (-1↑𝑁) ∈ ℂ)
41	25	nncnd 11683	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐻) ∈ ℂ)
42	40, 41, 40	mul32d 10881	. . . . . . . 8 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) = (((-1↑𝑁) · (-1↑𝑁)) · (2↑𝐻)))
43	18	nn0cnd 11989	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
44	43	2timesd 11910	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
45	44	eqcomd 2765	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 𝑁) = (2 · 𝑁))
46	45	oveq2d 7167	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝑁 + 𝑁)) = (-1↑(2 · 𝑁)))
47		neg1cn 11781	. . . . . . . . . . . 12 ⊢ -1 ∈ ℂ
48	47	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → -1 ∈ ℂ)
49	48, 18, 18	expaddd 13555	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁)))
50	18	nn0zd 12117	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
51		m1expeven 13519	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(2 · 𝑁)) = 1)
53	46, 49, 52	3eqtr3d 2802	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑𝑁) · (-1↑𝑁)) = 1)
54	53	oveq1d 7166	. . . . . . . 8 ⊢ (𝜑 → (((-1↑𝑁) · (-1↑𝑁)) · (2↑𝐻)) = (1 · (2↑𝐻)))
55	41	mulid2d 10690	. . . . . . . 8 ⊢ (𝜑 → (1 · (2↑𝐻)) = (2↑𝐻))
56	42, 54, 55	3eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) = (2↑𝐻))
57	56	oveq1d 7166	. . . . . 6 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((2↑𝐻) mod 𝑃))
58	40	mulid2d 10690	. . . . . . 7 ⊢ (𝜑 → (1 · (-1↑𝑁)) = (-1↑𝑁))
59	58	oveq1d 7166	. . . . . 6 ⊢ (𝜑 → ((1 · (-1↑𝑁)) mod 𝑃) = ((-1↑𝑁) mod 𝑃))
60	57, 59	eqeq12d 2775	. . . . 5 ⊢ (𝜑 → (((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃) ↔ ((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃)))
61	2	oveq2i 7162	. . . . . . . 8 ⊢ (2↑𝐻) = (2↑((𝑃 − 1) / 2))
62	61	oveq1i 7161	. . . . . . 7 ⊢ ((2↑𝐻) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃)
63	62	eqeq1i 2764	. . . . . 6 ⊢ (((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃) ↔ ((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃))
64		2z 12046	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
65		lgsvalmod 25992	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((2 /_L 𝑃) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃))
66	64, 1, 65	sylancr 591	. . . . . . . . 9 ⊢ (𝜑 → ((2 /_L 𝑃) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃))
67	66	eqcomd 2765	. . . . . . . 8 ⊢ (𝜑 → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((2 /_L 𝑃) mod 𝑃))
68	67	eqeq1d 2761	. . . . . . 7 ⊢ (𝜑 → (((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃) ↔ ((2 /_L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃)))
69	1, 4, 2, 5	gausslemma2dlem0i 26040	. . . . . . 7 ⊢ (𝜑 → (((2 /_L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
70	68, 69	sylbid 243	. . . . . 6 ⊢ (𝜑 → (((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
71	63, 70	syl5bi 245	. . . . 5 ⊢ (𝜑 → (((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
72	60, 71	sylbid 243	. . . 4 ⊢ (𝜑 → (((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
73	39, 72	syld 47	. . 3 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
74	16, 73	sylbid 243	. 2 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1 → (2 /_L 𝑃) = (-1↑𝑁)))
75	6, 74	mpd 15	1 ⊢ (𝜑 → (2 /_L 𝑃) = (-1↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ∖ cdif 3856 ifcif 4421 {csn 4523 class class class wbr 5033 ↦ cmpt 5113 ‘cfv 6336 (class class class)co 7151 ℂcc 10566 ℝcr 10567 1c1 10569 + caddc 10571 · cmul 10573 < clt 10706 − cmin 10901 -cneg 10902 / cdiv 11328 ℕcn 11667 2c2 11722 4c4 11724 ℕ₀cn0 11927 ℤcz 12013 ℝ⁺crp 12423 ...cfz 12932 ⌊cfl 13202 mod cmo 13279 ↑cexp 13472 ℙcprime 16060 /_L clgs 25970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5157 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-inf2 9130 ax-cnex 10624 ax-resscn 10625 ax-1cn 10626 ax-icn 10627 ax-addcl 10628 ax-addrcl 10629 ax-mulcl 10630 ax-mulrcl 10631 ax-mulcom 10632 ax-addass 10633 ax-mulass 10634 ax-distr 10635 ax-i2m1 10636 ax-1ne0 10637 ax-1rid 10638 ax-rnegex 10639 ax-rrecex 10640 ax-cnre 10641 ax-pre-lttri 10642 ax-pre-lttrn 10643 ax-pre-ltadd 10644 ax-pre-mulgt0 10645 ax-pre-sup 10646

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-pss 3878 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-tp 4528 df-op 4530 df-uni 4800 df-int 4840 df-iun 4886 df-br 5034 df-opab 5096 df-mpt 5114 df-tr 5140 df-id 5431 df-eprel 5436 df-po 5444 df-so 5445 df-fr 5484 df-se 5485 df-we 5486 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6127 df-ord 6173 df-on 6174 df-lim 6175 df-suc 6176 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-isom 6345 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-om 7581 df-1st 7694 df-2nd 7695 df-wrecs 7958 df-recs 8019 df-rdg 8057 df-1o 8113 df-2o 8114 df-oadd 8117 df-er 8300 df-map 8419 df-en 8529 df-dom 8530 df-sdom 8531 df-fin 8532 df-sup 8932 df-inf 8933 df-oi 9000 df-dju 9356 df-card 9394 df-pnf 10708 df-mnf 10709 df-xr 10710 df-ltxr 10711 df-le 10712 df-sub 10903 df-neg 10904 df-div 11329 df-nn 11668 df-2 11730 df-3 11731 df-4 11732 df-5 11733 df-6 11734 df-n0 11928 df-xnn0 12000 df-z 12014 df-uz 12276 df-q 12382 df-rp 12424 df-ioo 12776 df-fz 12933 df-fzo 13076 df-fl 13204 df-mod 13280 df-seq 13412 df-exp 13473 df-fac 13677 df-hash 13734 df-cj 14499 df-re 14500 df-im 14501 df-sqrt 14635 df-abs 14636 df-clim 14886 df-prod 15301 df-dvds 15649 df-gcd 15887 df-prm 16061 df-phi 16151 df-pc 16222 df-lgs 25971

This theorem is referenced by: 2lgs 26083

W3C validator